Trying to understand spacetime curvature

I'm familiar with space and time together being 4 dimensions and that mass causes a curvature in this spacetime.

When I consider a line that is curved, I can view the curvature because the line is drawn on a 2D surface (plane). So, it seems an additional dimension is required for a curvature to take place. I don't believe that you could curve a line in only one dimension, but I could be wrong (especially if curved could be thought of as moving backward and forward only, but this does not represent what we mean by a curved line when speaking commonly).

So, this brings me back to spacetime. Do the entire 4 dimensions curve? Or is it like my (perhaps nonsensical) line example where fewer than 4 dimenions curve and use the remaining dimension to move within? Or is it better to view each dimension as curving within another dimension, so that no extra dimension is needed to curve within? For instance, I could draw a cross (represents a 2D coordinate system) and move the lines around (x-axis curves using y-axis by moving up or down in a wavy way) on the paper but not lift them (as this would require adding a 3rd dimension to the scheme).

I'm likely confused by the common rubber sheet examples of spacetime, where an additional dimension is needed. This representation always left me wondering whether the space dimensions (if they can be thought of separately with any coherency) were curving within the time dimension, and is now making me question what it even means to curve.

This stuff IS difficult to "visualize"....it does not comport with our typical daily experiences. My ruler, for example, always seems the same length....no matter how I look at it.

Just you you see it: "gravity" is affected by things beyond mass....like energy and momentum.

"Do the entire 4 dimensions curve?"

Space and time ARE curved...better to think warped or distorted or changed.... in GR...in fact they are aspects of each other...they are related!! They are NOT independent entities as Netwon visualized!!! Two things change them: relative velocity and gravitational potential.

One way to think about this is inside a black hole...where r (distance) becomes time(t) and t becomes distance!!! This seems impossible to visualize. At the singulairty, space and time and mass seem to cease to exist as we know them...although we have no complete theory about that yet.

Another way to think about it is that gravitational potential curves both space and slows time as viewed from a distant inertial frame, So, for example, space on the surface of the earth is slightly curved, and as viewed from a distant point in free space, time appears to pass more slowly than it does locally at the distant observation point.

For time distortions...try eading about the effects of time delay for the GPS system in such wide use today...without time distortion corrections, that system would be useless in a few weeks...maybe a few days....

You can begin to see that space and time are related in unexpected ways relative to classical thinking...where space and time are each fixed, they do not vary....in special relativity where the Lorentz transformations shows space and time are related via velocity!!!

Finally, it may be that space and time are quantized, discrete, rather than continuous in quantum mechanics versus General Relativity. In some way, Planck scales (time,distance,energy) such as 10-33 cm MAY be about the minimum scale at which distance as we know it exists. And 10-43 sec MAY be about the minimum time "quanta"...

In it, I post an interesting explanation from DrGreg who explains different types of curvature due to acceleration on one hand and gravitational potential on the other...I did NOT mention acceleration in the prior post because it gets "deep in the forest" ...complicated.

Also as a sort of summary idea when DrGreg says "A geodesic has zero 4-acceleration and zero curvature." I believe he is explaining that whereas in flat space the shortest distance between two space points is a straight line, in spacetime, the shortest interval is a geodesic. So even "curvature" changes a bit in GR.

So, let me ask the group this in another way. Let's say I am only speaking of 3D space, but I want it to curve like 4D spacetime. Can I curve 3D space without needing a 4th dimension? If so, is similar to how I might view a glass cube with coordinates lines etch on the inside and outside being heated and smushed around? I suppose I am confusing curving a 3D object with curving a 3D coordinate system. I assume there are conditions that the coordinate system must possess even after being deformed.

So, let me ask the group this in another way. Let's say I am only speaking of 3D space, but I want it to curve like 4D spacetime. Can I curve 3D space without needing a 4th dimension? If so, is similar to how I might view a glass cube with coordinates lines etch on the inside and outside being heated and smushed around? I suppose I am confusing curving a 3D object with curving a 3D coordinate system. I assume there are conditions that the coordinate system must possess even after being deformed.

Yes, you can curve 3D space without a 4th dimension. The curvature is intrinsic and does not need to be embedded. Thinking about this curvature as curved axes is fine. The idea of curved and non-uniformly scaled axes is embodied in a metric, which defines an invariant infinitesimal measure of distance in the curved space.

If 3Ds can be curved, what role does the time "dimension" play in conceptualizing curved spacetime?

If you draw a largish parallelogram on a curved surface, you'll find that the opposite sides might not necessarily be equal in length, even though the sides are parallel.

Now, imagine such a parallelogram, but that one of the sides of this parallelogram is the time axis on a space-time diagram.

What you'll see, to make a long story short , is something that looks pretty much like gravitational time dilation. One timelike worldline will be shorter than another, even though
they are connected by parallel geodesics.

I have studied some of this stuff and sometimes it is easier just to use an equation and crunch a number. Sometimes understanding is much harder. If time is curved with space, is there some way to measure it? If the spacetime I live in is curved, is it anything that can be noticed?

If I take a piece of paper and bend it we would have a constant gravitational field but would we think it to be curvature? Technically only if I deform the paper do we have curvature.

You are right, that sort of curving doesn't count as "curvature" in the sense we use it in GR. If you can flatten something without stretching, squashing or tearing (e.g. the curved surface of part of a cylinder or cone) then it's not "curvature" in that sense.

"Curvature" is a rather vague word. To demonstrate space-time curvature in the stricter sense that Dr. Greg mentions, one needs to show the presence of geodesic deviation. If initially parallel geodesics do not remain a constant distance apart, space-time is curved in the strict sense of having a non-zero curvature tensor.

Gravitational time dilation, while it's a strong motivation for considering the presence of curvature, can be present in flat space-time if one uses non-inertial coordinates.

Staff: Mentor

I am reading the answers and would like to know what we exactly think curvature is.

If I take a piece of paper and bend it we would have a constant gravitational field but would we think it to be curvature? Technically only if I deform the paper do we have curvature.

Just checking if we are all talking about the same thing.

You can take a flat piece of paper and bend, fold, or roll it up. All of that is what is known as extrinsic curvature. One thing you cannot do is take a flat sheet of paper and roll it into a sphere. That is because a sphere has intrinsic curvature. Using only measures available within the 2D surface, a triangle will have exactly 180° on the paper, regardless of how it is bent in 3D, but a triangle will always have greater than 180° on the sphere. The curvature of GR is intrinsic curvature.

Thanks for all the comments, it really helps. I still will have to research further because this is very interesting to me. I started reading this lately in my Steven Hawking books: "the Universe in a Nutshell" and in "A Brief History of time". So I have one more mini-question, is Steven Hawking still the latest and greatest scientist in this field?